Question

The equations of two waves acting in perpendicular directions are given as $x=a\cos (\omega t+\delta )$ and $y=a\cos (\omega t+\alpha )$, where $\delta =\alpha +\frac{\pi }{2}$, the resultant wave represents

• A. a circle (c.w)
• B. a circle (a.c.w)
• C. an Ellipse (c.w)
• D. an ellipse (a.c.w)

Answer 1

Answer: (B)

Given : $x=a\cos (\omega t+\delta )$
and $y=a\cos (\omega t+\alpha )...(i)$
where, $\delta =\alpha +\pi /2$
$\therefore x=a\cos (\omega t+\alpha +\pi /2)$
$=-a\sin (\omega t+\alpha )...(ii)$
Given the two waves are acting in perpendicular direction with the same frequency and phase difference $\pi /2$
From equations (i) and (ii),
$x^2+y^2=a^2$
which represents the equation of a circle.